What does IOPS (in Amazon EBS) mean in practice?
As per your use case, s3 is a better option but if one wants to use an EBS volume and thinks that they require more IOPS, they can choose gp3 volume type instead of gp2. In gp3 volume, one can increase upto 16,000 IOPS independent of throughput (also, throughput can be increase upto 1000 MiB/s independently of IOPS).
You are missing the "/GB" part of that statement. The baseline is 3 IOPS per GB. If your EBS volume is 100GB, then you would have a baseline of 300 IOPS. For a GP2 EBS volume you have to multiple the size of the volume by 3 to get the IOPS.
Note that any GP2 volume under 1TB is also able to burst at up to 3,000 IOPS, so any limited increases in IO should still perform very well.
Also, I will add that S3 sounds like a better fit for your use case. If you are seeing slow upload speeds to S3, that is a problem that can be solved. You can use CloudFront to provide a nearby edge location that you can upload to.
In my experience uploads to S3 are never any slower than uploads to an EC2 instance that your EBS volume would be attached to.
Update:
To answer your additional question, the minimum IOPS needed will depend on many variables such as the amount of RAM available, the type of application you are running, how well the application caches values in memory, the average size of your IO operations, etc. It's really difficult to pin-down an exact number and state that you need exactly X IOPS for an application.
You also need to remember that any volume under 1TB in size can still burst up to 3,000 IOPS for several seconds. So even if your application needs high IOPS when it is in use, if it doesn't see much usage the IOPS burst feature might be all it ever needs.
In general I usually start with something like a 100GB volume with 300 IOPS and test the performance of my app against that. A web server that operates entirely within RAM might never need more than that. For something like a database you would probably start out with the amount of disk space you think you will need and then start performance testing. CloudWatch will show the amount of IOPS your application is using, and if you see it maxing out at the limits of your volume then you would know you need to increase the available IOPS. Rinse and repeat until you no longer max out the available IOPS during your performance tests.
@Mark B's answer is probably correct, in that it points out your IOPs are based on the size of your EBS volume. For what you want, S3 is the best option.
But depending on your use case and requirements, EBS may be needed. This is especially true if you want to run a database. In that case, you have a couple of options.
You can get Provisioned IOPS - if you know you need 5000 IOPS, but only need say 100GB of storage (which with gp2 would normally provide you with around 300 IOPS), you can use io1 volumes. There is an extra cost to this, and you'll want to make sure that it's attached to an EBS optimized instance, but you can get up to 20k IOPS if needed.
If you're doing a lot of sequential reads (reading in a large data set?) then there's a new type of EBS, st1. This is good for 500MB/s, and is less than 1/2 the cost of gp2.
Finally, there's one other scenario you could consider (say, you're a bit of a madman, and want to try doing strange things). If you can grab an archive from somewhere, and all you care about is serving them up from a really fast file system, you could put them on an instance that has instance storage. This is a locally-attached SSD, so it's very fast. The only drawback is that when your instance stops, you data is gone.
To address your update, "how many IOPS do you need for a database", the answer is "it depends". Every database engine has different requirements, and every database use has different usage patterns. Take a look at this if you want more information. But basically, test & monitor. If you're worried, over provision at launch, and scale down as needed. Or take a guess, and increase if you run into problems - is it more important to minimize costs, or provide good performance to your end users?